Pendulum Clock Length Calculator
Calculate the correct pendulum length for a clock based on desired beat period using the pendulum equation.
The period of a simple pendulum is governed by one of the most elegant equations in physics, first derived by Christiaan Huygens in 1673 in the Netherlands.
The Pendulum Period Formula
T = 2 x pi x sqrt(L / g)
Where:
- T = period of one complete swing (seconds)
- L = length from pivot to center of mass of the bob (meters)
- g = gravitational acceleration (9.80665 m/s^2 at sea level)
Solving for length: L = g x (T / (2 x pi))^2
Beat vs. Period
A “beat” in horology is one swing (half a period). A clock that “beats seconds” swings once per second, meaning the full period T = 2 seconds.
| Clock Type | Beats Per Minute | Beat Time | Period (T) | Pendulum Length |
|---|---|---|---|---|
| Seconds pendulum | 60 | 1.0 s | 2.0 s | 99.4 cm (39.1 in) |
| Vienna regulator | 60 | 1.0 s | 2.0 s | 99.4 cm |
| Grandfather clock | 60 | 1.0 s | 2.0 s | 99.4 cm |
| Mantel clock | 120 | 0.5 s | 1.0 s | 24.8 cm (9.8 in) |
| Wall cuckoo clock | 80 | 0.75 s | 1.5 s | 55.9 cm (22.0 in) |
| Small mantel | 160 | 0.375 s | 0.75 s | 14.0 cm (5.5 in) |
Worked Example — Grandfather Clock (Beats Seconds)
Beat time = 1.0 second, so period T = 2.0 seconds. L = 9.80665 x (2.0 / (2 x 3.14159))^2 L = 9.80665 x (0.31831)^2 L = 9.80665 x 0.10132 L = 0.9936 m = 99.36 cm
This is why the classic “seconds pendulum” is almost exactly one meter long.
Altitude and Latitude Corrections
Gravity varies slightly by location. At higher altitudes, g is smaller, so the pendulum must be shorter. At the equator, g = ~9.780 m/s^2. At the poles, g = ~9.832 m/s^2. A clock calibrated at sea level in London will run about 0.5 seconds slow per day if moved to Mexico City (elevation 2,240m).
Temperature Effects
A steel pendulum rod expands about 0.0012% per degree Celsius. For a 1-meter pendulum, a 10°C temperature increase lengthens the rod by about 0.12 mm. This causes the clock to lose roughly 0.5 seconds per day. This is why precision clocks use compensating pendulums made of invar alloy or wood-and-brass composite rods.
Practical Adjustment
Most pendulum clocks have a rating nut at the bottom of the bob. Turning the nut clockwise (raising the bob) shortens the effective length and speeds up the clock. One full turn of the rating nut typically changes timekeeping by 1–3 seconds per day.